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\frame{\frametitle{\r Replica Exchange Method and Expanded Ensembles}\b 

The late 1980s and early 1990s saw a flurry of innovative activities
on MCMC methods.
\bigskip \s

Swendsen and Wang, Phys. Rev. 58 (1987) 86-88; Wolff Phys. Ref.
Lett. 62 (1989) 361-363: \b Cluster algorithms.
\bigskip \s

Ferrenberg and Swendsen, Phys. Rev. Lett. 61 (1988) 2635-2638, 63 
(1989) 1658: \b Popularized {\r histogram reweighting} of canonical 
MCMC simulations and focused them on determinations of finite-Volume 
maxima of divergent quantities in studies of phase transitions.
\medskip \s

Ferrenberg and Swendsen, Phys. Rev. Lett. 63 (1989) 1195-1198;
Alves, Berg and Villanova, Phys. Rev. B 41 (1990) 383: \b
Multi- histogram reweighting.
\medskip \r

Question: \b Which of the earlier references relate to reweighting?
\bigskip \s
\vskip 20pt }


\frame{\frametitle{\r }\r 

Replica Exchange: \medskip \s

Swendsen and Wang, Phys. Rev. 57 (1986) 2607-2609: \b Introduced
(too) general replica exchange method.
\medskip \s

Frantz, Freemann, and Doll, J. Chem. Phys. 93 (1990) 2769-2784: \b
{\r J}ump{\r -walking} feeds replica from a high- into a low-temperature 
simulation. Does not fulfill balance (\r not \b replica exchange).
\medskip \s

Geyer, in Proceedings of the 23rd Symposium on the Interface, Keramidas 
(editor), Interface Foundation, Fairfax, Virginia (1991) 156-163: \b
Multiple Markov chains (the same as replica exchange).
\medskip \s

Hukusima and Nemoto, J. Phys. Soc. Japan 65 (196) 1604--1608: \b
Replica exchange. Often called \r parallel tempering \b when 
temperatures are exchanged.
\medskip \s

Hansmann, Chem. Phys. Lett. 281 (1997) 1267-1276: \b First application 
to a biomolecule (Met-Enkephalin).
\vskip 20pt }



\frame{\frametitle{} \bigskip\r 

Parallel Tempering: \bigskip \b

For $\s \beta_1 &lt; \beta_2 &lt; ... &lt; \beta_n$ consider the {\r joint 
partition function} 
\begin{equation}\s\nonumber 
  Z = \prod_{k=1}^n \left\lbrace \prod_{i=1}^N 
      \frac{(2\pi\,m_i\,k_B\,T_k)^{3/2} }{N!} 
      \int d^3 x_i^k\, \exp\left[ -\beta_k\,
      U(\vec{r}_1^{\,k},\dots,\vec{r}_N^{\,k})\right]
      \right\rbrace 
\end{equation}
of a molecular system, where the momenta are integrated out. {\r Swaps}
between (normally neighboring) temperatures are proposed and accepted 
with the Metropolis probability
\begin{equation}\s\nonumber 
  P_{\rm acpt} = \min \left\lbrace 1,\,\exp \left[+(\beta_k-\beta_l)\,
         \left( U(\vec{r}_1^{\,k},\dots,\vec{r}_N^{\,k}) -
                U(\vec{r}_1^{\,l},\dots,\vec{r}_N^{\,l})\right)
                \right]\right\rbrace
\end{equation}
so that {\r balance holds.} The differences $\s \beta_k-\beta_l$ have 
to be chosen small enough to ensure reasonable acceptance rates. 
\bigskip

\centerline{Well-suited for parallel processing.} \medskip

\centerline{Other advantages: To be discussed.} \medskip

\centerline{Generalized ensembles can be exchanged as well, ... }
\medskip
\vskip 20pt }


\frame{\frametitle{ }\b 

Do not confuse with \medskip \r

Expanded Ensembles: \bigskip \s

Lyubartsev, Martsinovski, Shevkanov, and Vorontsov- Velyaminov, J.~Chem.
Phys 96 (1992) 1776-1783. \medskip

Marinari and Parisi, Europhys. Lett. 19 (1992) 451-458: \b Simulated 
tempering. \medskip

\r Weights \s
\begin{equation} \nonumber
  \exp(-\beta_m\,E_k+g_m),~~m=1,\dots,M,~~k\ {\rm configuration,}
\end{equation} \b
and $\s m$ is considered a new dynamical variable. Moves
\begin{equation} \nonumber \s
  m \to m \pm 1
\end{equation} \b
are proposed and rejected/accepted according to Metropolis.
The constants $\s g_m$ need to be adjusted to get reasonable
acceptance rates.
\vskip 20pt }


\frame{\frametitle{ } \r
{\large How to overcome a free energy barrier?}
\begin{figure} \begin{center}
\includegraphics[width=0.55\textwidth]{figures/gauss2.pdf} 
\end{center} \end{figure}   
\begin{enumerate} \b
\item {\r Flatten the barrier.} This requires that the barrier is {\r 
      explicit} in a known reaction coordinate. Probabilities of 
      transition states are then greatly enhanced.
\item {\r Jump the barrier} via parallel or simulated tempering. 
      {\r Initial low-temperature configuration(s)} problematic! 
      Probabilities of transition states stay unchanged (tiny). 
      Parallel tempering will not estimate entropy and free energy 
      across the barrier.
% Simulated tempering does. Question: Why?
\end{enumerate}
\vskip 20pt }

\frame{\frametitle{ } \r
Exploration of low temperature configurations in a rugged free energy 
landscape:
\begin{figure} \begin{center}
\includegraphics[width=0.6\textwidth]{figures/rugged.pdf} 
\end{center} \end{figure}  \b 
If there are no major barriers in the energy variable, all of the 
introduced methods explore all low temperature branches and yield
their relative probabilities as well as the global energy minimum. 
\medskip

Due to {\r barriers in \s $M$:} Large performance enhancement versus 
low temperature Boltzmann simulations.
\vskip 20pt }

\frame{\frametitle{ } \b
Example (left over from my last MCMC course at FSU): \medskip

{\r Parallel tempering} with 8 processes for the 2D 10-state Potts 
model on a $\s 20\times 20$ lattice. Some parameters: \smallskip \s

\centerline{$h_q=0$, $\beta_{\min}=0.65$, $\beta_{\max}=0.75$,
            (note $\beta_t=0.71303\dots)$.}

\begin{table}
% \caption {\label{table1} } \medskip
\begin{tabular}{|c|c|c|c|c|} \hline
&amp;\multicolumn{2}{c|}{Parallel Tempering}&amp; 
\multicolumn{2}{c|}{Canonical Simulation} \\ \hline
$\beta$  &amp;$\tau_{\rm int}(E)$&amp;$\tau_{\rm int}(M)$
         &amp;$\tau_{\rm int}(E)$&amp;$\tau_{\rm int}(M)$ \\ \hline
0.7133...&amp;  $761\,(21)$&amp;$434\,(13)\times 10$
         &amp; $1199\,(51)$&amp;$181\,(10)\times 10^2$ \\ \hline
0.7261...&amp;$54.0\,(4.9)$&amp; $611\,(19)\times 10$ 
         &amp;$13.63\,(22)$&amp; $979\,(48)\times 10^4$\\ \hline
\end{tabular}
\end{table} \b

Units are sweeps. Assembled statistics: \s $10\times 2^{20}$ per
replica for PT, $10\times 2^{20}$ cano at $\beta=0.7133...$, 
$10^3\times 2^{22}$ cano at $\beta=0.7261...$~. \bigskip \b

{\r Exercise BB3:} Explain why the integrated autocorrelation times 
{\s $\tau_{\rm int}(E)$ and $\tau_{\rm int}(M)$} behave so differently.
\medskip

% Hints: (1)~Do not confuse autocorrelation times and integrated 
% autocorrelation times. (2)~You have to guess additional information
% about the runs.
\vskip 20pt }

\frame{\frametitle{\r Summary and Conclusions: }\b 

This talk presented use of generalized ensembles in MCMC simulations 
close to the {\r chronological order} of their development. Obviously, 
my mini-review cannot be complete, I had to make omissions and may have 
overlooked some relevant papers.
\bigskip

A message is that mastering simulation methods is {\r not} the 
trivial part of a biophysical, chemical or physical MCMC study. 
{\r Astronomically large efficiency factors} can float around 
between making it right or wrong.
\bigskip

There have been hundreds of papers refining and improving the methods 
outlined here. It is practically impossible to follow up on all of 
them, all the more to verify them. Let me just pick four points:
\vskip 20pt }


\frame{\frametitle{}\b  \vskip 10pt

\begin{enumerate}

\item Wang-Landau, PRL 86 (2001) 361-363: Introduced a now widely used 
algorithm, which can be employed as a recursion to get working estimates 
of the MUCA weights.

\item Trebst, Huse and Troyer, PRE 70 (2004) 046710: Optimized the
MUCA weight with respect to cycling times. Improvement factors
appear to be around 3 to 5 compared to relying on flat histograms.

\item \r Hamiltonian replica exchange, alchemical transitions \b 
and related ideas have been studies by a number of authors, including 
Uli Hansmann and Wei Yang, who are here and may tell you more. \s

\item Data analysis problems should not be neglected. For instance,
reweighting often requires a logarithmic coding, because factors just
get too large to be allowed in Fortran or~C. That is a good occasion
to advertise my book for your library:

\end{enumerate} 
\vskip 20pt }

\frame{
\begin{figure} \begin{center}
\includegraphics[width=1.0\textwidth]{figures/MC3.jpg} 
\end{center} \end{figure}   }


\frame{\frametitle{}\b 

Really {\r huge improvement} will get attention, because they would be 
{\r enabling methods} for simulations which otherwise cannot be done. 
However, attention is not enough. Required are independent verifications,
which too often never happen.

% Though, there is the Hatano-Gubernatis danger, \s PRB 66, 054437 
% (2002), \b a paper which reported MUCA autocorrelation 
% times $\s \sim V$ in an EAI spin glass simulation, which to the day 
% has not been reproduced independently.
\bigskip

{\r Where are we stuck?}
\bigskip

Apparently at many fronts, but we focus here on the simulation of
well-defined models (all-atom in biophysics, classical spin glasses 
in statistical physics, and so on).

\begin{enumerate}

\item Obviously there are often hidden barriers and we are unable to 
find {\r reaction coordinates} in which the barriers become explicit.

\item The optimal performance of the discussed methods is limited by
a {\r diffusive process.} For large systems that can still be far too 
slow. In some situations that is greatly improved by the collective
updating of cluster algorithms, but their application range has
remained fairly limited.

\end{enumerate} 

\r Challenges: \b Many. Combinations with MD simulations as for instance
pioneered by Wei Yang appear to me one of the promising roads.

\medskip \vskip 20pt }
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